VGAM (version 1.0-6)

simplex: Simplex Distribution Family Function

Description

The two parameters of the univariate standard simplex distribution are estimated by full maximum likelihood estimation.

Usage

simplex(lmu = "logit", lsigma = "loge", imu = NULL, isigma = NULL,
        imethod = 1, ishrinkage = 0.95, zero = "sigma")

Arguments

lmu, lsigma

Link function for mu and sigma. See Links for more choices.

imu, isigma

Optional initial values for mu and sigma. A NULL means a value is obtained internally.

imethod, ishrinkage, zero

See CommonVGAMffArguments for information.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Details

The probability density function can be written $$f(y;\mu ,\sigma )=[2\pi {\sigma }^2(y(1-y){)}^3{]}^{-0.5}\exp [-0.5(y-\mu {)}^2/({\sigma }^{2}y(1-y){\mu }^2(1-\mu {)}^2)]$$ for $0<y<1$, $0<\mu <1$, and $\sigma >0$. The mean of $Y$ is $\mu$ (called mu, and returned as the fitted values).

The second parameter, sigma, of this standard simplex distribution is known as the dispersion parameter. The unit variance function is $V(\mu )={\mu }^3(1-\mu {)}^3$. Fisher scoring is applied to both parameters.

References

Jorgensen, B. (1997) The Theory of Dispersion Models. London: Chapman & Hall

Song, P. X.-K. (2007) Correlated Data Analysis: Modeling, Analytics, and Applications. Springer.

See Also

dsimplex, dirichlet, rig, binomialff.

Examples

# NOT RUN {
sdata <- data.frame(x2 = runif(nn <- 1000))
sdata <- transform(sdata, eta1 = 1 + 2 * x2,
                          eta2 = 1 - 2 * x2)
sdata <- transform(sdata, y = rsimplex(nn, mu = logit(eta1, inverse = TRUE),
                                       dispersion = exp(eta2)))
(fit <- vglm(y ~ x2, simplex(zero = NULL), data = sdata, trace = TRUE))
coef(fit, matrix = TRUE)
summary(fit)
# }